Physics 2102
Jonathan Dowling
Physics 2102
Lecture 2
Electric Fields
January 17, 07
Version: 1/17/07
Charles-Augustin
de Coulomb
(1736-1806)
What are we going to learn?
A road map
• Electric charge
 Electric force on other electric charges
 Electric field, and electric potential
• Moving electric charges : current
• Electronic circuit components: batteries, resistors, capacitors
• Electric currents  Magnetic field
 Magnetic force on moving charges
• Time-varying magnetic field  Electric Field
• More circuit components: inductors.
• Electromagnetic waves  light waves
• Geometrical Optics (light rays).
• Physical optics (light waves)
Coulomb’s law
+ q1
F12
F21
r12
k | q1 | | q2 |
| F12 |=
2
r12
!q2
For charges in a
VACUUM
2
N
m
9
8
.
99
!
10
k=
C2
Often, we write k as:
k= 1
4$# 0
with # 0 = 8.85 " 10
!12
2
C
N m2
Electric Fields
• Electric field E at some point in
space is defined as the force
experienced by an imaginary
point charge of +1 C, divided by
1 C.
• Note that E is a VECTOR.
• Since E is the force per unit
charge, it is measured in units of
N/C.
• We measure the electric field
using very small “test charges”,
and dividing the measured force
by the magnitude of the charge.
Electric field of a point charge
+1 C
q
E
R
k |q|
| E |= 2
R
Superposition
• Question: How do we figure out the field
due to several point charges?
• Answer: consider one charge at a time,
calculate the field (a vector!) produced by
each charge, and then add all the vectors!
(“superposition”)
• Useful to look out for SYMMETRY to
simplify calculations!
Example
Total electric field
-2q
+q
• 4 charges are placed at the
corners of a square as shown.
• What is the direction of the
electric field at the center of
the square?
-q
y
+2q
(a) Field is ZERO!
(b) Along +y
(c) Along +x
x
Electric Field Lines
• Field lines: useful way to
visualize electric field E
• Field lines start at a
positive charge, end at
negative charge
• E at any point in space is
tangential to field line
• Field lines are closer
where E is stronger
Example: a negative point
charge — note spherical
symmetry
Electric Field of a Dipole
• Electric dipole: two point
charges +q and –q separated
by a distance d
• Common arrangement in
Nature: molecules, antennae,
…
• Note axial or cylindrical
symmetry
• Define “dipole moment”
vector p: from –q to +q, with
magnitude qd
Cancer, Cisplatin and electric dipoles:
http://chemcases.com/cisplat/cisplat01.htm
Electric Field ON axis of dipole
-q
a
+q
P
x
Superposition : E = E+ + E!
E+ =
kq
a#
&
$x' !
2"
%
2
&
#
$
!
1
1
!
E = kq $
2
2
$,
a) ,
a) !
$* x - ' * x + ' !
2( +
2( "
%+
E' = '
= kq
kq
a#
&
$x+ !
2"
%
2 xa
2 #2
& 2 a
$x ' !
$
!
4
%
"
2
Electric Field ON axis of dipole
E = kq
2 xa
2 #2
& 2 a
$x ' !
$
!
4
%
"
=
p = qa
“dipole moment”
-- VECTOR
2kpx
2 #2
& 2 a
$x ' !
$
!
4
%
"
-
+
What if x>> a? (i.e. very far away)
2kpx 2kp
E! 4 = 3
x
x
r
E!
r
p
r
3
E~p/r3 is actually true for ANY point far from a dipole
(not just on axis)
Electric Dipole in a Uniform Field
• Net force on dipole = 0;
center of mass stays where
it is.
• Net TORQUE τ: INTO
page. Dipole rotates to line
up in direction of E.
• | τ | = 2(QE)(d/2)(sin θ)
= (Qd)(E)sinθ
= |p| E sinθ
= |p x E|
• The dipole tends to “align”
itself with the field lines.
• What happens if the field is
NOT UNIFORM??
Distance between charges = d
Electric charges and fields
We work with two different kinds of problems, easily confused:
• Given certain electric charges, we calculate the electric field
produced by those charges
(using E=kqr/r3 for each charge)
Example: the electric field produced
by a single charge, or by a dipole:
• Given an electric field, we calculate the forces applied by this
electric field on charges that come into the field, using F=qE
Examples: forces on a single charge
when immersed in the field of a dipole,
torque on a dipole when immersed in
an uniform electric field.
Continuous Charge Distribution
• Thus far, we have only dealt
with discrete, point charges.
• Imagine instead that a charge
Q is smeared out over a:
Q
Q
– LINE
– AREA
– VOLUME
• How to compute the electric
field E??
Q
Q
Charge Density
λ = Q/L
• Useful idea: charge density
• Line of charge:
charge per unit length = λ
• Sheet of charge:
charge per unit area = σ
• Volume of charge:
charge per unit volume = ρ
σ = Q/A
ρ = Q/V
Computing electric field
of continuous charge distribution
• Approach: divide the continuous
charge distribution into
infinitesimally small elements
• Treat each element as a POINT
charge & compute its electric
field
• Sum (integrate) over all elements
• Always look for symmetry to
simplify life!
Example: Field on Bisector of Charged Rod
• Uniform line of charge +Q
spread over length L
• What is the direction of the
electric field at a point P on
the perpendicular bisector?
(a) Field is 0.
(b) Along +y
(c) Along +x
• Choose symmetrically
located elements of length dx
• x components of E cancel
P
y
a
x
dx
o
L
dx
q
Example --Line of Charge: Quantitative
• Uniform line of charge,
length L, total charge Q
• Compute explicitly the
magnitude of E at point P
on perpendicular bisector
• Showed earlier that the net
field at P is in the y
direction -- let’s now
compute this!
P
y
a
x
o
L
Q
Line Of Charge: Field on bisector
Distance
dE
P
Charge per unit length
dx
x o
L
q
!=
L
k (dq )
dE =
2
d
a
d!
d = a2 + x2
Q
k (! dx)a
dE y = dE cos" = 2
(a + x 2 )3 / 2
a
cos! = 2
2 1/ 2
(a + x )
Line Of Charge: Field on bisector
L/2
L/2
dx
'
$
x
E y = k# a !
2
2 3 / 2 = k( a % 2
2
2"
(
a
+
x
)
& a x + a # !L / 2
"L / 2
=
2k!L
2
2
a 4a + L
What is E very far away from the line (L<<a)?
What is E if the line is infinitely long (L >> a)?
2k!L
2k!
Ey =
=
2
a
a L
Example -- Arc of Charge: Quantitative
• Figure shows a uniformly
charged rod of charge −Q bent
into a circular arc of radius R,
centered at (0,0).
• Compute the direction &
magnitude of E at the origin.
kdQ
dE x = dE cos! = 2 cos!
R
Ex =
" /2
!
0
k ($Rd# ) cos# k$
=
2
R
R
k!
Ex =
R
y
450
x
y
dQ = λRdθ
dθ
" /2
! cos#d#
0
k!
k!
Ey =
E= 2
R
R
θ
x
λ = 2Q/(πR)
Example : Field on Axis of Charged Disk
• A uniformly charged circular
disk (with positive charge)
• What is the direction of E at
point P on the axis?
z
(a) Field is 0
(b) Along +z
(c) Somewhere in the x-y plane
P
y
x
Example : Arc of Charge
y
• Figure shows a uniformly
charged rod of charge -Q
bent into a circular arc of
radius R, centered at (0,0).
• What is the direction of the
electric field at the origin?
x
(a) Field is 0.
• Choose symmetric elements
(b) Along +y
• x components cancel
(c) Along -y
Summary
• The electric field produced by a system of charges at
any point in space is the force per unit charge they
produce at that point.
• We can draw field lines to visualize the electric field
produced by electric charges.
• Electric field of a point charge: E=kq/r2
• Electric field of a dipole:
E~kp/r3
• An electric dipole in an electric field rotates to align
itself with the field.
• Use CALCULUS to find E-field from a continuous
charge distribution.